The aim of this book is to explain modern homotopy theory in a manner
accessible to graduate students yet structured so that experts can skip over
numerous linear developments to quickly reach the topics of their interest.
Homotopy theory arises from choosing a class of maps, called weak equivalences,
and then passing to the homotopy category by localizing with respect to the
weak equivalences, i.e., by creating a new category in which the weak
equivalences are isomorphisms. Quillen defined a model category to be a
category together with a class of weak equivalences and additional structure
useful for describing the homotopy category in terms of the original category.
This allows you to make constructions analogous to those used to study the
homotopy theory of topological spaces.

A model category has a class of maps called weak equivalences plus two other
classes of maps, called cofibrations and fibrations. Quillen's axioms ensure
that the homotopy category exists and that the cofibrations and fibrations
have extension and lifting properties similar to those of cofibration and
fibration maps of topological spaces. During the past several decades the
language of model categories has become standard in many areas of algebraic
topology, and it is increasingly being used in other fields where homotopy
theoretic ideas are becoming important, including modern algebraic
$K$-theory and algebraic geometry.

All these subjects and more are discussed in the book, beginning with the
basic definitions and giving complete arguments in order to make the
motivations and proofs accessible to the novice. The book is intended for
graduate students and research mathematicians working in homotopy theory and
related areas.